EXCHANGE 


MAK  16  1316 


CONTRIBUTIONS  TO 
EQUILONG  GEOMETRY 


BY 


PAUL  HENRY  LINEHAN 


Submitted    in  Partial    Fulfillment  of    the  Requirements   for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1915 


CONTRIBUTIONS  TO 
EQUILONG  GEOMETRY 


BT 

PAUL  HENRY  LINEHAN 


Submitted   in  Partial    Fulfillment  of   the  Requirements   for 

THE  Degree  of  Doctor  of  Philosophy,  in  the  Faculty 

OF  Pure  Science,  Columbia  University 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 

1915 


U^' 


§ 

a: 
c 


CONTENTS. 

Page 

Introduction v 

Chapter  I. 

The  Invariants  of  Irregular  Analytic  Curves  under  the  Group  of  Equilong 

Transformations  of  the  Plane 1 

Chapter  II, 

Some  Curves  with  Equations  in  Hessian  Line  Coordinates. 

Sect.  1.  Hessian  Line  Coordinates 4 

2.  Evolutes  and  Involutes 5 

3.  Radii  of  Curvature 8 

4.  Pedals  and  Negative  Pedals 9 

Chapter  III. 
Linear  Equilong  Transformations. 

Sect.  1.  The  Integral  Linear  Transformations 12 

(a)  The  Operation  W  =  w  +  ^ 12 

(6)  The  Operation  W  =  aw 15 

(c)  The  Operation  W  =  aw  +  ^ 19 

Sect.  2.  The  General  Linear  Transformations 21 

(o)  The  Operation  W  =  - 21 

w 

(6)  Combinations  of  the  Operations  W  =  w  +  B  and  W  =  —.  25 

w 

(c)  Combinations  of  the  Operations  W  =  aw  ■{-  B  and  W  =  —.  27 

w 

(d)  The  Fixed  Lines  and  the  Criteria  for  the  Types  of  Motion.  31 

(e)  The  Simplest  Invariant  of  a  Regular  Analytic  Curve 35 


3^- 


Si' 


7802 


111 


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http://www.archive.org/details/contributionstoeOOIinerich 


INTRODUCTION. 

The  infinite  group  of  equilong  transformations  of  the  plane*  is  that  group 
of  contact  transformations  of  the  plane  by  which  a  straight  line  becomes  a 
straight  line  and  distance  on  the  line  is  unchanged.  Two  arbitrary  lineal 
elements  of  one  and  the  same  line  become,  under  an  equilong  transformation, 
two  lineal  elements  of  the  transformed  line,  the  distance  between  the  points 
of  the  original  elements  being  equal  to  the  distance  between  the  points  of  the 
new  elements. 

If  the  lines  of  the  plane  be  determined  by  the  Hessian  coordinates  {u,  v), 
V  being  the  length  of  a  perpendicular  from  a  fixed  point  to  the  line  and  u  being 
the  angle  between  the  perpendicular  and  a  fixed  line  through  the  fixed  point,t 
and  if  the  lines  of  the  transformed  plane  be  determined  by  the  Hessian  co- 
ordinates {U,  V),  the  infinite  group  of  equilong  transformations  will  be 
represented  by 

U-\-jV  =  f(u+jv), 

f{u  +  jv)  denoting  an  arbitrary  analytic  function  of  the  so-called  dual  number 
u  +  jv  of  a  system  with  the  unities  1  and  j  in  which  j^  =  0. 
The  equilong  transformations  may  be  expressed  by 

U  =  <p{u), 

d(p{u)  , 

in  which  (p  and  x  are  arbitrary  analytic  functions  of  u. 

In  this  article  several  topics  in  equilong  geometry  are  treated. 

Chapter  I  is  concerned  with  the  existence  and  derivation  of  the  invariants 
of  irregular  analytic  curves  under  the  group  of  equilong  transformations  of 
the  plane.  Except  for  certain  types,  invariants  always  exist.  The  results 
are  roughly  analogous  to  Kasner's  results  for  the  conformal  group. { 

In  Chapter  II,  the  use  of  Hessian  line  coordinates  in  the  determination  of  a 
lineal  element  and  in  the  equations  of  curves  is  first  discussed.  Then,  by 
means  of  these  coordinates,  the  equations  of  the  nth  evolute  and  the  nth 

*  Cf .  Scheffers,  Isogonalkurven,  Aquitangentialkurven,  und  komplexe  Zahlen,  Mathe- 
matische  Annalen,  bd.  60. 

t  Hessian  line  coordinates  are  completely  described  in  Chap.  II,  1. 

X  Cf.  Kasner,  Conformal  classification  of  analytic  arcs  or  elements;  Poincar^'s  local  prob- 
lem of  conformal  geometry,  Trans.  Amer.  Math.  Soc,  vol.  16  (1915),  pp.  333-349, 


VI  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

involute  are  derived.  Expressions  for  the  successive  radii  of  curvature  of  an 
analytic  curve  in  terms  of  the  coeflBcients  of  the  infinite  series  representing  the 
curve  are  also  obtained.  Finally  it  is  shown  that  if  an  equation  be  interpreted 
successively  in  Hessian  line  and  polar  point  coordinates  the  equation  represents 
a  curve  and  its  pedal  respectively. 

In  Chapter  III,  the  equilong  transformations  of  the  plane  which  are  repre- 
sented by  the  linear  functions  of  the  dual  variable  u  +  jv  are  studied.  The 
treatment  is  largely  after  the  manner  of  treatment  of  the  conformal  trans- 
formation represented  by  the  linear  functions  of  the  complex  variable  x  +  iy* 
It  is  seen  that  all  linear  transformations  consist  of  certain  particular  trans- 
formations and  their  combinations.  For  the  several  types  of  motion,  the 
effect  on  certain  systems  of  curves,  the  systems  of  curves  along  which  the 
motions  may  be  assumed  to  take  place,  the  invariants,  the  fixed  lines,  and  the 
conditions  are  obtained.  The  simplest  invariant  of  the  regular  analytic  curve 
under  the  general  linear  transformation  is  obtained  also. 

The  writer  acknowledges  with  great  pleasure  his  indebtedness  to  Professor 
Kasner  for  helpful  suggestions  and  criticisms. 

*  Cf .  Cole,  The  Linear  Functions  of  a  Complex  Variable,  Annals  of  Mathematics,  vol.  5. 


CHAPTER  I. 

THE    INVARIANTS     OF    IRREGULAR    ANALYTIC    CURVES     UNDER    THE 
GROUP  OF  EQUILONG  TRANSFORMATIONS  OF  THE  PLANE. 

If  the  irregular  analytic  curve* 

V  =  agW«/P  +  a;,+iw(«+i>/P  +  ag+2W<«+2)/P  ^  ... 
is  to  go  into  the  irregular  analytic  curve 

V  =  A^U^I^  +  A^+iU^^+'^lP  +  A^+^U^^'^lP  4-  •  •  •, 

p  and  q  being  positive  integers,  p  >  2  and  q  >  p,  under  the  general  equilong 
transformation 

U  =  z2  o^ry^, 

r=l 

CO  «8 

V  =zl  vrarvr~^  +  zZ  bsW, 

r=l  «=1 

then  the  result  obtained  by  eliminating  U,  V,  and  v  among  the  four  equations 
must  be  an  identity  in  u  or 


E  E  ra^5+;fcwf«+('-i>^*^/^  +  Z  6.W*  -  E  ^ 


3+n 
n=0 


^  +  n  H-"  (1  —  r)p 


=  0; 


"^rtl  a:!2/!2!... 

X,  y,  z,  •  • '  being  positive  integers  or  zero  satisfying  the  relation 

x-\-y-\-z-\----^r. 

Suppose  p  =  2.  If  the  coefficients  of  any  number  of  the  powers  of  u  be 
each  equated  to  zero,  the  number  of  the  constants  of  the  transformation 
present  in  the  equations  will  always  be  found  to  equal  or  to  exceed  the  number 
of  equations.     Hence  there  are  no  invariants. 

*  An  irregular  analytic  curve  (arc)  is  a  curve  which,  in  the  neighborhood  'of  a  given  Une 
taken  as  the  line  (0,  0)  of  the  Hessian  coordinate  system,  cannot  be  represented  by  setting  v 
equal  to  a  series  in  integral  powers  in  u  but  can  be  represented  by  a  series  with  fractional 
exponents. 

1 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


Suppose  p  >  2.    First,  if  neither  q/p  nor  (q  +  l)/p  be  an  integer  then  from 
the  equations  formed  by  equating  to  zero  the  coeflScients  of  w'/p  and  u^^^^l^, 

aiUq  —  tti^l^Ag  =  0, 

aia^+i  -  ai<«+«/M5+i  =  0, 

the  constant  oi  of  the  transformation  may  be  ehminated  and  an  invariant 

obtained. 

Secondly,  if  qfp  be  an  integer,  (q  -{-  l)fp  and  {q  +  2)fp  will  not  be  integers. 

From  the  equations  formed  by  equating  to  zero  the  coeflBcients  of  w(«+i)/i» 
andi*(«+2)/p^ 

the  constant  ai  may  be  eliminated  and  an  invariant 

.9+2-p 


«9+r 


obtained. 


«g+2 


9+i-p 


=  J 


fl+2 


Fig.  1. 

Thirdly,  if  {q  +  l)lp  be  an  integer,  q(p  and  (q  +  2)/p  will  not  be  integers. 
Then  equating  to  zero  the  coeflBcients  of  u^'^  and  u^^^'>Ip  gives  the  two 

equations 

aiaq  —  ai«/Mg  =  0, 

aia^i  -  ai(«+2)/M,+2  =  0, 
from  which  oi  may  be  eliminated  and  the  invariant 


a 


«+2-p 


obtained. 


a  9+2* 


=  J 


fl+2 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  3 

Hence  every  irregular  analytic  curve,  except  the  curves 

has  equilong  invariants.* 

Of  the  types  which  have  no  invariants,  the  form  for  q  =  S, 

is  represented  in  Fig.  1. 

*  Regular  curves  are  all  reducible  to  the  normal  form  v  =  0  and  hence  have  no  equilong 
invariants.  Cf.  Kasner,  Conformal  Geometry,  Proc.  Fifth  Inter.  Congress  of  Math.,  Cam- 
bridge, 1912,  vol.  2,  p.  85. 


CHAPTER  II. 
SOME  CURVES  WITH  EQUATIONS  IN  HESSIAN  LINE  COORDINATES. 

1.   Hessian  Line  Coordinates. 

A  plane  curve  is  to  be  regarded  as  the  envelope  of  a  single  infinity  of 
oriented  straight  lines  each  of  which  is  determined  by  the  two  Hessian  co- 
ordinates (u,  v),  V  being  the  length  of  a  perpendicular  from  a  fixed  point  to 
the  line  (w,  v)  and  u  being  the  angle  between  the  perpendicular  (v)  and  a 
fixed  line  through  the  fixed  point.  On  the  perpendicular  (v),  the  distance 
from  the  fixed  point  to  the  line  {u,  v)  is  considered  positive  or  negative  according 
as  the  fixed  point  is  on  the  left  or  on  the  right  of  the  direction  of  the  line  (w,  v). 
The  angle  (u)  is  generated  by  rotating  the  fixed  line  until  its  positive  sense 
coincides  with  the  positive  sense  of  the  perpendicular  (v)  and  is  positive  or 
negative  according  as  the  rotation  is  in  the  counter-clockwise  or  in  the  clock- 
wise direction. 

A  lineal  element  of  the  plane  is  ordinarily  determined  by  (x,  y,  dy/dx), 
{x,  y)  being  the  cartesian  coordinates  of  its  point  and  dyjdx  the  tangent  of  the 
angle  which  its  direction  makes  with  the  positive  direction  of  the  ic-axis.  It 
may  also  be  determined  by  the  quantities  (w,  v,  dv/du),  {u,  v)  being  the  Hessian 
coordinates  of  its  line  (direction)  and  dv/du  the  distance  along  its  line  from 
the  foot  of  the  perpendicular  (v)  to  its  point.  If  (x,  y,  dy/dx)  and  (w,  v,  dv/du) 
represent  the  same  lineal  element,  the  following  relations  exist:* 

u  =  tan  ^-3 ^  , 

ax      2 

dy 

""d^-y 

V  = 


Mtr 


du 


>Ri)^' 


In  general  a  plane  curve  is  oriented  and  is  assumed  to  have  an  equation 
of  the  form 

*  Cf.  Scheffere,  loc.  cit.,  p.  520. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  5 

V  =  f{u)* 

The  equation  u  =  k  (a,  constant)  represents  a  point  at  infinity  which  is,  of 
course,  regarded  as  a  curve. 

It  is  easily  seen  geometrically  that  the  contact  transformation 


U=^ 

u  +  ^u 

V  = 

«, 

dV 
dU~ 

dv 

is  simply  rotation  of  the  plane  about  the  fixed  point  «  =  0  through  an  angle  j8i. 
It  is  clear  also,  geometrically,  that  the  transformation 

dV_dv 
dU~  du' 

is  dilatation  of  the  plane  through  the  distance  ^.  In  general,  then,  a  curve, 
involute  of  some  curve,  E,  remains  an  involute  of  that  same  curve,  E,  under 
the  transformation 

V=  v-\-^. 

2.    E VOLUTES  AND   INVOLUTES. 

The  nth  evolute  and  the  nth  involute  of  a  curve,  v  =  f(u),  may  be  easily 
obtained. 

Let  the  curve  v  =  /(w),  the  evolute  of  which  is  sought,  be  tangent  to  the 
line  AB(u,  v)  (Fig.  2)  at  B.  BD,  which  is  perpendicular  to  AB,  will  be  tangent 
to  the  evolute  at  D.    Let  the  coordinates  of  BD  be  (wi,  Vi).    Then 


Fig.  2. 
*  This  form  is  easily  seen  to  be  similar  to  the  better  known  "magical  equation  of  the 
tangent  to  a  curve."     Cf.  Loria,  Sp.  Alg.  u.  Trans.  Ebene  Kurven  (Gr.  trans,  by  Schiitte), 
bd.  II,  p.  255. 


6  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

0C=  AB 
or 

dv      df(u) 

Replacing  «  by  Wi  —  ^  » 

is  obtained.     Hence  the  equation  of  the  first  evolute  of  the  curve 

V  =fiu) 


IS 


'=/'(«-!)• 


Since,  however,  the  substitution  of  w  +  r  for  u  is  equivalent  simply  to 

rotating  a  curve  through  the  angle  —  7r/2  about  the  point  »  =  0  of  the  plane, 
the  equation  of  the  first  evolute  of  the  curve 

V  =  f(u) 
may  be  taken  to  be 

It  follows  that  the  nth  evolute  of  a  curve 

V  =  f{u) 
is 

|,=/(n)(w), 

where 


/("Hw) 


du"* 


To  place  the  evolute  in  its  natural  position  with  respect  to  the  curve,  it  should 
be  rotated  through  the  angle  n7r/2  or  its  equivalent. 

The  equation  of  the  involute  of  a  curve  may  be  deduced  independently  of 
the  results  just  obtained.  It  may  also  be  simply  derived  by  means  of  those 
results.     If  the  equation  of  the  involute  of  the  curve 

V  =  f{u) 
is  assumed  to  be 

V  =  g(u), 
then 

g'{u)=f{u) 
and 

^(w)  =  J  f{u)du  +  constant. 


CONTKIBUTIONS  TO  EQUILONG  GEOMETRY.  7 

Hence  the  equation  of  a  first  involute  of  the  curve 

«  =  f{u) 
is 

V  =  J  f(u)du  +  constant. 

The  variation  of  the  constant  through  all  possible  values  gives  the  oo^ 
involutes  of  the  curve.  In  particular,  the  constant  may  take  the  value  zero. 
Consequently  it  follows  that  the  first  involute  of  the  curve     ^ 

V  =  f{u) 
may  be  written 

V  =  J  f{u)du 
and  the  n  involute  may  be  written 

/(n)  

/(w)dw''. 

If  the  constant  is  not  taken  equal  to  zero  for  each  successive  involute, 
the  equation  of  the  nth  involute  is 

/in)  

the  c's  being  constants. 

To  place  the  nth  involute  in  its  natural  positive  with  respect  to  the  curve 
it  should  be  rotated  through  the  angle  Smr/2  or  its  equivalent. 

The  results  just  obtained  may  be  used  to  find  the  involutes  of  a  circle. 
If  the  equation  of  a  circle  be 

V  =  a, 

then  the  equation  of  the  first  involute  of  a  circle  is 

V  =  au  -\-  b 
and  of  the  second  involute 

«  =  r  n^  +  6m  +  c 

and  of  the  nth  involute 

»  =  ^""  +  (S^!"""'  +  (^!"""  +•••  +  *• 
Hence  an  equation  of  the  form 

V  =  CnW  +  Cn-iM**"^  +    •  •  •  +  Cq 

represents  the  nth  involute  of  the  circle 

V  =  Cn-nl 


8  contributions  to  equilong  geometry. 

3.  Radii  of  Curvature. 

The  radii  of  curvature  of  the  successive  evolutes  of  a  curve  (analytic  arc) 
at  a  tangent  of  the  curve  are  expressible  conveniently  in  terms  of  the  coeflS- 
cients  of  the  infinite  series  representing  the  arc,  and  conversely.  Let  the 
radius  of  curvature  of  the  kth.  evolute  be  denoted  by  r^,  the  radius  of  curvature 
of  the  curve  being  Tq.    Since 

ds  dH 

then 

_  dfQ  _d^s  _dv      dh 

^      du      du^      du      du^ 
and 

_  dH      d^h 

An  analytic  arc  is  represented  by  the  series 

U  =  Co  +  Ci2*  +  C2U^  +    •  •  •   +  CmV^  +    *  *  ', 

the  c's  being  constants.  By  rotation  and  translation*  the  arc  may  be  so 
placed  that  the  point  of  tangency  is  the  fixed  point  and  the  normal  through 
the  point  of  tangency  is  the  fixed  line  of  the  system  of  coordinates.  Accord- 
ingly, an  analytic  arc  may  be  represented  by  the  infinite  series 

ia  =  CiV?  +  CzU^  -\-  CiU^  +  •  •  •  +  CmU"^  +  •  •  • . 

At  the  line  (0,  0), 

dv 


du"- 


-r-5=  2c2, 


and 

Hence 


d'^v 
dvT 

d^v 
dh 
d  V      d     V 

*  Translation    {X  =  x  +  Pi]  Y  =  y  +  02',  dY/dX  =  dy/dx)   is  expressed  by  the  equa- 
tions U  =  u;  y  =  t>  +  ft  cos  u  +  ft  sin  u;  dV/dU  =  dv/du  —  ft  sin  u  +  ft  cos  u. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

If  these  equations  be  solved  for  the  c's,  then 


d^l, 

C2  = 

2!  " 

21' 

C3  = 

dh 

du^ 

3! 

d'^ 

~3!' 

Ci  = 

du* 

4! 

ra  -  fo 
41 

» 

Cf,  = 

d^v 

dw' 

'   5! 

rz—  Ti 
51 

» 

d^+h 

p 

E    (- 

l)'^V^2-2p 

and 


^"+'~  (n+2)!  (n  +  2)I 

in  which  p  takes  positive  integral  values. 

By  means  of  these  results,  the  Cauchy  test  for  the  convergeney  of  series 
may  be  interpreted  geometrically.     If  the  series 

V  =  C^W^  +   •  •  •  +  CkU^  +   •  •  • 

represent  an  analytic  arc,  then  the  superior  limit,  as  7i  =  oo ,  of 


n+2 


p  <  (nl2)  +  1 

i;    (-  i)^v,H-2-2p 
p=i 


(n+2)! 

is  finite.  If  this  condition  be  not  fulfilled  then  the  series  is  divergent  and 
represents  geometrically  not  an  analytic  curve  but  what  Kasner*  has  called  a 
"divergent  differential  element  of  infinite  order." 

4.  Pedals  and  Negative  Pedals. 

Suppose  the  equation  of  an  oriented  curve  be  expressed  in  polar  coordinates 
(p,  6).  Let  the  coordinates  be  slightly  modified  by  the  following  conditions. 
Let  the  direction  on  the  radius  vector  (p)  from  the  pole  to  a  point  of  a  curve 
be  positive  or  negative  according  as  the  pole  lies  on  the  left  or  on  the  right 
of  the  positive  direction  of  the  curve  at  the  point.  Let  the  conditions  which 
govern  the  sense  of  u  also  govern  the  sense  of  the  vectorial  angle  6.  Let  the 
positive  direction  on  the  polar  subnormal  (dp/dd)  be  that  which  makes  with 

*  Cf .  Bull,  of  the  Am.  Math.  Soc,  vol.  XX,  no.  10,  p.  531. 


10  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

the  positive  sense  of  the  radius  vector  (p)  an  angle  of  +  7r/2.  The  three 
magnitudes  (p,  6,  dp/dd)  may  be  used  to  determine  a  lineal  element  of  the 
plane.  The  point  of  the  element  is  fixed  by  (p,  6).  The  direction  of  the 
element  is  determined  by  dp/dd,  being  the  direction  perpendicular  to  the  line 
(normal  of  the  arc  of  the  curve)  joining  the  point  (p,  6)  to  the  extremity 
(other  than  the  pole)  of  the  polar  subnormal  and  being  determined  in  sense 
by  the  convention  determining  the  sense  of  the  radius  vector  (p). 
Consider  now  the  transformation 

e  =  u, 

dp      dv 
dd      du' 

This  is  seen  geometrically  (Fig.  3)  to  be  the  foot  point  transformation,  the 
element  {u,  v,  dv/du)  becoming  the  element  (6,  p,  dp/dd).  The  result  is  ob- 
tained analytically  by  expressing  the  transformation  in  the  coordinates 
{x,  y,  dy/dx),  representing  the  lineal  element  (w,  v,  dv/du),  and  {X,  Y,  dY/dX), 
representing  the  lineal  element  (6,  p,  dp/dd).  In  this  notation,  the  transforma- 
tion is 

dy 

X V 

i^^  ,    .,.„  dx      ^  ..Y  ,dy      TT 

VZ2  +  y2  =     ,         _  ^^ ,        tan-i  ^  =  tan-i:r  -  -  , 

X  dx      2 


dy 
dx 


^§-y 


>R1)^' 


or 

dy  f    dy         \  dy 


GI-) 


x=±^^  &    y       j,^^    dx 


^^m'     ^H-dr 


dY        "{dxj       *      ■'^dx 


dX 


m-^^Hty 


which  represents  the  foot  point  transformation  or  the  foot  point  transforma- 
tion followed  by  rotation  through  the  angle  tt,  according  as  the  upper  or  lower 
signs  are  taken  respectively. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


11 


Hence  if  a  curve  be  expressed  by  the  equation  v  =  f{u),  in  Hessian  line 
coordinates,  its  pedal,  with  respect  to  the  point  »  =  0,  is  expressed  by  the 
same  equation  p  =  f{6),  in  polar  point  coordinates.     Conversely,  if  a  curve 


Fig.  3. 

be  expressed  by  the  equation  p  =  f(d),  in  polar  point  coordinates,  its  first 
negative  pedal  with  respect  to  the  point  p  =  0,  is  expressed  by  the  same 
equation,  v  =  f(u),  in  Hessian  line  coordinates.  ' 

For  example,  the  equation 

P  =  Cn^**  +  Cn^l^"-!  +    •  •  •   +  Co 

represents  the  pedal  of  the  nth  involute  (v  =  c„w**  +  c„_iw"~i  +  •  •  •  +  Co) 
of  a  circle  (v  =  w!c„)  with  respect  to  the  center  of  the  circle  and  the  equation 
»  =  a"  represents  the  first  negative  pedal  of  the  equiangular  spiral  (p  =  a^) 
with  respect  to  its  pole  and  represents,  consequently,  an  equiangular  spiral. 

The  beauty  of  Scheffers'  treatment  of  equilong  transformations  is  due  to  the  combining 
of  the  Hessian  line  coordinates  with  the  dual  number  u  +  jv  (j^  =  0).  If  polar  coordinates 
were  combined  with  this  dual  number,  an  infinite  group  of  point  transformations  would 
be  obtained  under  which  the  difference  of  the  polar  subnormals  (with  respect  to  a  fixed  point) 
of  two  concurrent  curves  at  their  common  point  would  be  invariant. 


CHAPTER  III. 

LINEAR  EQUILONG  TRANSFORMATIONS. 

1.  The  Integral  Linear  Transformations. 

(a)  The  Operation  W  =  w  +  /3. 
The  integral  linear  function,  W  =  U  -\-  jV,  of  the  dual  variable, 

w  =  u  -\-  jv, 
is 

W  =  aw  +  P, 

where  a  and  /3  are  dual  numbers  of  the  same  form  as  W  and  w. 
If  a  =  1,  the  function  becomes 

W=  w  +  fi, 

which  represents  the  transformation 

U  +  jV=  u  +  jv  +  Pi  +  j^ 
or 

f^  =  w  +  /3i, 

V  =  v+^, 

dV_dv^ 
dU      du ' 

First,  let  /3i  4=  0  and  /^  =  0.  Replacing  the  coordinates  (w,  r,  dvjdu  =  v'), 
which  may  be  referred  to  as  the  Hessian  coordinates  of  a  lineal  element,  by 
the  coordinates  (x,  y,  dyjdx  =  y'),  cartesian  coordinates,  the  transformation 
takes  the  form 

tan-ir-|=tan-i2/'-|  +  ^i, 

XY'  -  Y       xy'  -y 
^1  +  Y'^ ~  ^l  +  y'^' 
X+YY'  ^x^yy^ 

or,  solving  for  X,  Y,  and  Y', 

*  The  equilong  transformations  may  be  regarded  as  transformations  of  lines  or  of  lineal 
elements  in  the  plane. 

12 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  13 

X  =  X  COS  Pi  —  y  sin  fii, 
Y  =  X  smPi-{-  y  cos  /3i, 
y'  +  tan  /3i 


Y'  = 


1  —  y'  '  tan  /3i  * 


These  equations  define  rotation  through  the  angle  j8i  about  the  point  v  =  0. 
Every  curve  v  ==  k,  a,  circle  with  center  at  this  point,  is  unchanged  as  a  whole 
although  its  elements  move  in  order  around  it.*  The  circles  v  =  k  are  there- 
fore the  curves  of  motion. f 

Suppose,  secondly,  that  jSi  =  0  and  P2  4=  0.     The  transformation  becomes 

U=u, 

V  =  v  +  ^, 


dU      du' 


In  cartesian  coordinates,  it  is 


tan  ^  y   —  —  =  tan     y   —  ^ , 

XY'-Y       xy'-y 

~r  P2, 


or 


Vi_|_  y/2 

<\^y"' 

Z+  FF' 

x-\-  yy' 

Vi  4-  y'2 

Vl  +  2,'^' 

Z  =  a;  + 

Piy' 

F  =  i/- 

/S2 

^1  +  .'^' 

1^'  =  y'. 

which  defines  dilatation. 

The  orientation  of  the  Hessian  coordinates  shows  that  the  dilatation  is  in 
the  direction  which  makes  with  the  positive  direction  of  the  lineal  element  an 

*  An  equilong  transformation  always  turns  every  curve  m  =  &,  a  point  at  infinity,  into 
a  curve  «  =  c,  another  (or  sometimes  the  same)  point  at  infinity. 

t  The  particular  curve  along  which  a  particular  line  may  be  regarded  as  moving  may  be 
obtained  by  substituting  the  coordinates  of  the  line  in  the  equation  of  the  family  of  curves, 
solving  for  the  parameter,  and  then  substituting  the  value  obtained  in  the  equation  of  the 
family  of  curves.     If  a  line  move  from  the  position  (a,  6)  to  the  position  (A,  B)  along  the 

curve  V  —  /(u),  the  distance  passed  over  is   I     [/(«)  +/"(u)]dw. 


14  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

angle  of  T  7r/2  according  as  ^  is  positive  or  negative.     Each  circle,  v  =  k, 
with  center  at  the  point  v  =  0,  becomes  another  circle,  v  =  k -{•  P2,  with  center 
at  the  same  point.    The  curves  of  motion  are  the  points  at  infinity,  u  =  k. 
The  transformation 

dU~du' 

in  which  j3i  =|=  0  and  182  4=  0,  is  rotation  and  dilatation  in  succession.    Since 
dv/du  is  unchanged  by  the  transformation,  the  00 1  systems  of  curves 

dv  _ 
du 

are  each  unchanged.     K  the  individual  curves  of  one  of  the  systems 

V  =  CU+  k 
remain  unchanged,  then,  after  the  substitution 

u=  U-fii, 

every  curve 

V=  cU-c^i  +  k  +  ^ 

must  be  identical  with  every  curve 

V  =  cu-\-  k. 


If  this  be  so, 
or 

Hence  the  curves 


-cPi-\-k-}-^  =  k 

ft 

^  =  «"• 
Pi 

ft      I    I 
v  =  —u+k, 

Pi 


the  00*  first  involutes  of  the  circle  v  =  ft/ft,  are  each  unchanged  as  a  whole 
and  may  be  regarded  as  the  curves  along  which  the  lines  move  under  the 
transformation. 

Since  dv/du  is  invariant  and  since 

d  _d_    dU  _d^ 

du~  dU'  du~  dU      ' 
or 

±_d_ 
du~  dU' 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  15 

every  derivative  d'^vldu^  is  invariant.  Consequently  the  radius  of  curvature 
of  every  evolute  of  a  curve  is  invariant.  An  involute  of  a  curve  is  transformed 
therefore  into  another  involute  of  the  same  curve.  For  rotation,  one  invariant 
is 

and,  for  dilatation, 

dv      dh 

du      du^       ^ 

which  last   is   also  an   invariant  of   the  rotation-dilatation   transformation 

(&)   The  Operation  W  =  aw. 
If,  in  the  integral  linear  function,  i3  =  0,  the  transformation 

W  =  U  -\-  jV  =  aw  =  (ai  +  ja2){u  +  jv) 
or 

U  =  aiU, 

V  =  aiV  +  a2U, 

dV  _  dv      a2 
dU      du      ai 
is  obtained. 

First,  let  0:2  =  0.    The  transformation  is  then 

U  =  aiU, 

V  =  aiV, 

dV_di 
dU      du ' 

The  line  (0,  0)  is  fixed.  A  circle  v  =  k,  with  center  at  the  point  v  =  0,  goes 
into  a  circle  v  =  aik,  with  center  at  the  same  point.     Since 


every  curve 


is  unchanged  as  a  whole.  Hence  the  curves  of  motion  are  v  =  Jcu,  those  00* 
first  involutes  of  the  00  ^  circles  v  =  k,  which  are  all  tangent  to  the  fixed  line 
(0,  0). 

*  It  is  evident  that  Hessian  line  coordinates  are  convenient  for  the  treatment  of  rotation 
and  dilatation. 


V 

u 

aiV 
aiu 

= 

V 

V 

u 

k 

16  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

Secondly,  let  ai  =  1.    The  transformation  is  then 

U=u, 
V  =  V  +  ttiU, 

dU~du'^''" 

The  points  at  infinity,  u  =  k,  are  the  curves  of  motion. 

If,  thirdly,  ai  4=  1  and  a2  4=  0,  every  circle  v  =  k,  with  center  at  the  point 
e  =  0,  is  transformed  into  a  first  involute,  v  =  {az/aiju  +  aik,  of  the  circle 

Let  the  equation  of  the  curves  along  which  the  lines  may  be  regarded  as 

moving  be 

V  =  f{u). 

By  the  transformation  this  equation  must  assume  the  form 

Hence  the  equations 

V  =  /(w) 
and 

aiv  +  «2W  =  fipciu) 

must  be  identical  or  the  function  /  must  satisfy  the  equation 

«i/(w)  +  «2W  =  /(aiw). 
Differentiation  gives 

/'(w)-/'(«iw)+^=0. 

Differentiation  a  second  time  gives 

j"{u)  -  a,j"{a,u)  =  0, 
or 

r'{u) 


and  differentiation  a  third  time, 

f"'{u)  -  <xiY"{cciu)  =  0. 
By  eliminating  the  coefficient  ai  between  the  last  two  equations,  the  equation 

f'"{u)        r"{aiu) 

is  obtained. 
Hence 

S"\u) 

,.,,,  v,»  =  a  constant. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


17 


The  solution  of  this  differential  equation,  obtained  by  direct  integration  three 
times  in  succession,  is 

/(w)  ==  a(u  +  6)  log  (w  +  6)  +  cw  +  d. 

The  equation  of  the  curves  is  then 

V  =  a{u  +  b)  log  (w  +  6)  +  cw  +  d, 

in  which  the  values  of  a,  h,  c,  and  d  are  to  be  determined.     If  this  equation  is 
to  be  unchanged  in  form  by  the  transformation,  then 


Hence  the  oo '  curves  along  which  the  lines  may  be  regarded  as  moving  under 
the  transformation  are  represented  by  the  equation 

V  =  — i — i —  u  •  log  U-+-  cu. 
ailoglai 


ASYMPTOTE 


Fig.  4. 


Consider  the  curve  (Fig.  4) 


V  = 


«2 


ai log  ai 


w-log  w  +  cu 


or 


ai  log  ai  » 


It  is  of  spiral  shape  since  v  increases  as  u  increases.     If 

M  =  0 


18 
then 
since 
and 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 
V=  0 

[w-Iogw]„=o  =  0 


00. 


[-1     = 

The  line  (0,  0)  is  an  asymptote. 

The  first  evolute  of  this  logarithmic*  curve  is 


V  = 


«2 


log 


(«-^)  +  (o  +  ^^), 


Q!i  log  ai         V  2  /       \         ai  log  ai 

which  is  the  first  negative  pedal  with  respect  to  the  point  «  =  0  of  the  logarith- 
mic spiral  of  Varignon,  the  equation  of  which  is,  in  polar  coordinates, 

^  =  -(^n^)'''«(''-|)-(-"-in^)- 

The  second  evolute  is 


v{u  —  x)  = 


"2 


ai  log  ai ' 
which  is  the  first  negative  pedal  of  the  hyperbolic  spiral 

ailogai 
with  respect  to  its  pole. 

From  the  equations  of  the  transformation,  it  follows  that 

A-      A 
du  ~  "'  dU 


and  hence  that 


du-       "' 


dU- 


[n  =  2,  3,  4,  .  • .]. 


Therefore  every  expression  of  the  form 


is  invariant,  the  simplest  being 


_du''  _ 

m-l 

~drv  ' 
_du"*_ 

n-1 

[m  =  2,  3,  4,  •  •  •  4=  n] 


du^j 

dh 

du^ 


=    ^3 


*  The  curve  «  =  a(«  +  b)  log  (u  +  6)  +  cu  +  d> — o,  b,  c,  and  d  being  constants, — ^will 
be  referred  to  as  a  "logarithmic  curve." 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  19 

The  invariants  containing  derivatives  of  even  order  only  may  be  expressed 
in  terms  of  the  radii  of  curvature  of  the  curve  and  its  evolutes  of  even  order. 
Consider  an  analytic  curve  in  the  neighborhood  of  the  line  (0,  0).    Then 


and 


du'-''' 

dH 
du'  -  '' 

-  ro, 

9=1 

l)'^V2p-2,. 

[p=  1,2,3,  •••]. 


Consequently  the  invariants  are 


2p+l 


[^2p+2^  -|2p-l  r  q=p+\  ~|2p-l' 


The  invariant  of  fourth  order 


may  be  written 


Idu^l 

dS 
du* 

To' 

r2—  tq' 


(c)   The  Operation  W  =  aw  -{-  fi. 
The  general  integral  linear  function 

W  =  aw  +  fi 
represents  the  transformation 

U  =  aiU-{-  jSi, 

V  =  aiv  -j-  a^u  +  jS2, 

dV  _  dn      az 
dU      du      ai ' 

Every  circle  v  =  k,  with  center  at  the  point  v  =  0,  goes  into  the  curve 

«2      ,    /  ^       a2iSi  ,       ,  \ 

a  first  involute  of  the  circle  v  =  a2fai. 


20  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


.  = -,=^  [«  -  j^]  log  [«  -  j4^]  +  0« 


By  the  method  employed  in  the  case  of  the  transformation  W  =  aw,  the 
system  of  curves  along  which  the  lines  may  be  regarded  as  moving  is  found  to 
be 

ai  log  ai 

■  «2/3i  +  caifii  —  cjSi  —  ai^  +  ^ 

a  system  of  logarithmic  curves  with  the  line 

/     /3i         (1  -Q!i)/32  4-«2/3i\ 

Vl-ai'  (l-«i)'         J 

as  asymptote.* 

This  system  of  curves  may  be  obtained  from  the  system  of  curves 

Vl  =    ^— Ml -log  Ml  -\-  CMi, 

ai log  ai 
for  the  transformation 

Wi  =  aiDi, 

by  the  rotation-dilatation  transformation 

w  =  M?i  +  ;■ . 

1  —  a 

Hence  the  transformation 

W  =  aw  +  fi 

may  be  regarded  geometrically  as  the  transformation 

Wi  —  aw\, 
where 

W\  =  w  —  -T- — — 
1  —  a 

and 

1  —  a 

or  in  which  the  fixed  line  of  the  transformation  is 

j8  /     jSi         (l-«i)/32  +  «2/gi\ 

l-«°'"Vl-ai*  (l-«i)'         /• 

The  invariants  of  the  transformation  expressed  by  the  general  integral 
linear  function  PF  =  aw;  +  j8  are  those  which  are  common  to  the  transforma- 
tions expressed  by  the  particular  integral  linear  functions  W  =  w  -{-  fi  and 

*  If  ai  =  1,  the  system  of  curves  is 
and,  if  /9i  =0  also,  the  system  b  «  =  A. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  21 

W  =  aw  and  are,  consequently,  those  of  the  latter.    The  invariant  of  lowest 
order  is 


dh 


=  Jz. 


=  —  a 


The  system  of  oo '  logarithmic  curves 

Idu'i 

dh 
du^ 
or 

V  =  a{u  +  6)  log  (u  -{-  b)  -{-  cu  -\-  d, 

b,  e,  and  d  being  arbitrary,  is  invariant. 

The  equilong  property  of  the  transformation  is  easily  deduced  from 

dV  _dv      a2 
dU      du      ai' 

If  two  curves,  tangent  to  the  line  (u,  v),  have  the  distances  [dv/du]i  and  [dvldu]^ 
respectively,  their  transformed  curves,  tangent  to  the  line  {U,  V),  will  have 
the  distances 

du  Ji      ai 
and 


fdVl  _  Vd^l 
ldUii~ldu], 

vdvi  _r^] 


2         «1 

respectively,  and  consequently 

Idui      lduii~ldU]2      IdUJi 

and  the  distance  between  the  points  of  tangency  is  invariant. 

2.  The  General  Linear  Transformations. 

(a)   The  Operation  PF  =  —  . 
The  reciprocal  function 

W  =  -* 

w 

represents  the  transformation 

*  The  lines  u  +  jv  where  u  =  0  are  not  considered.     Throughout  the  entire  article,  the 
concept  of  a  line  or  of  lines  at  infinity  is  not  considered  and  division  hyj  is,  of  course,  eliminated. 


22  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

1  u  —  jv  u  —  jv 


U  +  jV  = 


or 


u  -\-  jv      {u  +  jv)  (u  —  jv)  u^ 

u 


dV  _  dv  V 

dU      du         u 


A  circle  v  =  k,  with  center  at  the  point  «  =  0,  is  transformed  into  the 
curve  V  =  —  ku^  which  is  a  second  involute  of  the  circle  v  =  —  2k  and  which 
is  tangent  to  the  line  (0,  0). 

Since,  from  the  equations  of  the  transformation, 


the  system  of  oo  i  curves 

or 

is  invariant.     Each  curve 
goes  into  the  curve 
the  curve 


do^_v_dV_V 
du      u~  dU      U' 


dv       ^  _  7 
du      u 


V  =  ku-  log  u  -\-  cu 

V  =  ku- log  u-\-  cu 
V  =  ku- log  u  —  cu, 

V  =  ku-  log  u 


going  into  itself. 

Hence  the  oo  ^  logarithmic  curves 

V  =  ku-  log  u 

are  the  curves  along  which  the  lines  may  be  regarded  as  moving  in  order. 
The  transformation 

U  +  jV  =  -^ 

u  +  JV 
is  equivalent  to  the  reverse  equilong  transformation 

U  +  jV  =  ^-^ 

U  —  JV 

or 

U 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  23 

V 

V  =  — 

dU~  ~  du^    u' 

followed  by  the  reverse  equilong  transformation 

U  -{-  jV  =  u  —  jv 
or 

U=u, 

V  =  -V, 

dV^  _dv 

dU  du ' 

This  last  transformation  is  equilong  symmetry  with  respect  to  the  point 
V  =  0.  Equilong  symmetry  with  respect  to  the  curve  v  =  f{u)  is  defined  in 
general  by  the  equations* 

V  =  -v+  2/(w), 

dV _  _dv      ^dfju) 
dU  du  du 

In  cartesian  coordinates,  the  transformation  in  this  special  case  is 

X=  -X, 

dY_dy 
dX     dx' 

which  is  rotation  about  the  point  v  =  0  through  the  angle  t.  Since  a  lineal 
element  and  its  symmetrical  element  are  pointed  in  the  same  direction,  the 
transformation  has  the  effect  of  rotating  every  lineal  element  through  the 
angle  w  and  then  reversing  its  direction.  Every  curve  is  rotated  through  an 
angle  ir  and  has  its  orientation  reversed. 
By  the  transformation 

u 

V 

V  =  — 

dV  _       dv         V 
dU~  ~du'^u' 

*  This  definition  was  given  by  Kasner,  in  a  course  at  Columbia  University,  as  a  kind  of 
analogue  to  conformal  or  Schwarzian  symmetry. 


24  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

a  circle  v  =  k  goes  into  a  curve  v  =  ku^,  which  is  a  second  involute  of  a  circle 
V  =  2k  and  which  is  tangent  to  the  line  (0,  0).     Every  curve 

au-\-  hv  =  0, 

which  is  a  first  involute  of  a  circle  v  =  —  a/b  and  which  is  tangent  to  the  line 
(0,  0),  is  invariant.     A  first  involute,  not  tangent  to  (0,  0), 

au  -\-  hv  -\-  c  =  0 

is  transformed  into  a  second  involute  (spiral  of  Sturm  or  Norwich) 

cu^  -j-  bv  -\-  au  =  0, 

tangent  to  (0,  0).    A  second  involute,  tangent  to  (0,  0), 

cu^  -{-  bv  -{-  au  =  0 
goes  into  a  first  involute 

au  •\-  bv  -\-  c  =  0, 

not  tangent  to  (0,  0).    A  second  involute,  not  tangent  to  (0,  0), 

cu^  -\-  au  •}-  bv  +  d  =  0 
is  transformed  into  a  second  involute 

du^  +  aw  +  6r  +  c  =  0, 

not  tangent  to  (0,  0).     Each  of  the  oo^  second  involutes 

cu^  +  aw  +  6«  +  c  =  0 
is  invariant. 

By  the  transformation,  the  two  points  at  infinity  w  =  db  1  are  rigid. 
Every  line  (±  1,  A;)  is  fixed.    A  line 

|wl  <1. 

V  =  k 
becomes  a  line 

\U\>h 

V>k 
and  a  line 

|w|>l, 

V  =  k 
becomes  a  line 

\U\  <1, 

V<k. 

The  lines,  of  course,  do  not  move  in  order  along  the  curves  of  any  system. 

The  transformation  is  of  period  2  and  is  its  own  inverse  as  is  also  the  trans- 
formation W  =  Ifw. 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  25 

In  the  reciprocal  transformation,  W  —  Ifw, 

dv      ^  _   T 
du      u        ^ 
is  invariant.    Since 

""du'       ^dU' 
the  invariant  of  order  n  may  be  written 

[  L    «w  J       \_du       w  J  j 

in  which  the  symbol  [w(d/dw)]"~^  signifies  the  repetition  (n  —  1)  times  of  the 
operation  u{d/du). 

For  the  reciprocal  transformation,  the  equilong  property  is  easily  shown  to 
exist.  If  (w,  V,  [dvfdu]i)  and  (u,  v,  [dvldu]2)  be  the  coordinates  of  the  lineal 
elements  of  two  curves  tangent  to  the  line  (m,  v)  and  if  {U,  V,  [dVldU]i)  and 
{U,  V,  [dV/dU]2)  be  the  corresponding  elements  for  the  transformed  curves 
tangent  to  the  line  {U,  V),  then 


and 

and  hence 


IdUJi      U~ldui2     u 

r^i  _z_  [—1  _- 

ldUJ2         ldUii~lduJ2         Idui 


(6)  Combinations  of  the  Operations  W  =  w  -\-  ^  and  W  =  — . 

If  the  lines  (wo)  of  a  plane  move  along  the  curves,  first  involutes  of  a  circle 

Vo  =  j82//3i, 

^  1      7 

Pi 

the  corresponding  lines  (w  =  Ijwo)  of  a  second  plane,  reciprocal  to  the  first, 
will  move  along  the  curves 

V  =  —  ku^  —  ttu- 
Pi 

These  are  second  involutes  of  the  circles  v  =  —  2k  and  are  tangent  to  the  line 
(0,  0)  at  the  same  point  {[dvldu]u=o  =  —  P2IP1)' 

The  equation  of  the  transformation  corresponding  to  motion  along  this 
system  of  curves  may  be  determined  in  the  following  way.    Let  Wo  and  w  be 


26  CONTRIBUTIONS  TO  EQUILONG  GEOMETRT. 

the  initial  positions  and  Wq  and  W  the  final  positions  of  lines  in  the  first  plane 
and  in  the  reciprocal  plane  respectively.  Since  the  motion  in  the  first  plane 
corresponds  to  the  transformation 

and  since 
and 


then  the  motion  in  the  reciprocal  plane  will  be  that  of  the  transformation 

or 

w 


Wo== 

Wq-\-^ 

w 

_  1 

Wo 

W-- 

1 

w 


^w-\-\ 


If  (in  /3  =  /3i  +  j^  /3i  =  0,  the  motion  is  along  the  points  at  infinity  u  =  l/k^ 
which  are  reciprocal  to  the  points  at  infinity  Uq  =  k. 

Suppose  the  system  of  second  involutes  to  have  the  line  (71,  72)  instead  of 
the  line  (0,  0)  in  common.     The  equation  will  then  be 

«>  —  72  =  —  kiu  —  7i)2  —  —  (m  —  71). 

Pi 

Subject  the  plane  to  the  transformation 

Wi  =  w  —  y. 
The  equation  of  the  system  of  second  involutes  of  circles  takes  the  form 

Vi=   —  ku^  —  -^  Ml, 

Pi 

the  line  (0,  0)  being  common  to  all  the  curves.    Since  motion  along  this 
system  corresponds  to  the  transformation 


in  which  wi  and  Wi  represent  respectively  the  initial  and  final  positions  of  the 

lines,  and  since 

wi  =  w  —  y 
and 

Wi=W-y, 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  27 

the  transformation 

or 

(1  4-  Mw  -  ^y" 


W  = 


Pw-  {I-  ^y) 


is  obtained.  This  Hnear  function  then  represents  motion  along  a  system  of 
curves 

»  -  72  =  —  k{u  —  7i)2  —  ^{u  —  7i), 

Pi 

each  curve  being  a  second  involute  of  a  circle  {v  =  —  2k)  and  being  tangent  to 
the  line  (71,  72)  at  the  same  point  ([dvldu]u=y  =  —  ^l^i).  Motion  of  this 
kind  will  be  called  parabolic. 

If  both  numerator  and  denominator  of  the  second  member  of  the  equation 

_  (1  +  ^y)w  -  ^7^ 
^w  +  (1  -  ^7) 

be  multiplied  by  an  arbitrary  number  5  and  if 

5  +  i857  =  a' 
and 

^5  =  7' 
and 

6  -  j857  =  3', 

then,  dropping  accents,  the  transformation  will  be  expressed  by  the  function 

(«  -  ^Y 

aw ;, 


yw  -\-  b 

(c)  Combinations  of  the  Operations  W  =  aw  -\r  ^  and  W  =  —. 

In  the  transformation 

W\  =  awi, 

if  (in  a  =  ai  +  jag)  (Xz  =  0,  the  curves  of  motion  are  Vi  =  kui,  which  are 
first  involutes  of  the  circles  v  =  k  and  which  have  the  line  (0,  0)  in  common. 
If  these  curves  be  subjected  to  the  transformation 

Wq=   Wi  +  71, 

their  equation  will  take  the  form 

«o  —  T/2  =  k{uQ  —  771), 
(vu  V2)  becoming  the  common  hne.     If  lines  (wq)  move  along  this  system  of 


28  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

curves,  the  reciprocal  lines  (w  =  1/wo)  will  move  along  the  system  of  curves 

V  =  {krji  —  r}2)u^  —  ku. 

Motion  along  these  curves,  which  are  second  involutes  of  the  circles 

V  =  2(krii  —  ife) 

and  which  have  the  lines  (0,  0)  and  (I/771,  —  772/771^)  in  common,  will  be  called 
hyperboHc. 

If,  in  the  transformation  {Wi  =  awi),  ai  =  1,  the  curves  of  motion  are  the 
points  at  infinity  ui  =  k.  The  lines  (w  =  Ifwi)  reciprocal  to  the  lines  moving 
along  these  points  will  move  along  the  points  at  infinity  u  =  Ifk. 

For  the  general  transformation 

Wi  =  awi, 

the  curves  along  which  the  lines  may  be  regarded  as  moving  are  the  00  * 
logarithmic  curves,  c  being  an  arbitrary  constant, 

Vi  =   — -, Wi-logMi  +  CUi. 

ai  log  ai 
If  these  curves  be  subjected  to  the  transformation 

they  become  the  00 1  logarithmic  curves 

^0-  V2=  ^  1  „^  (wo  -  Vi)  log  {uo  -  vi)  +  c(wo  -  Vi)* 
Qfi  log  ai 

the  reciprocal  curves  of  which  are,  since  w  —  1/wq,  the  qo^  curves 

V  =  ^^  I      ^^  [Viu"^  -  u]  [log  (1  -  riiu)  -  log  u]-  cu-\-  ^ct/i  -  rjzW- 

The  line  (0,  0)  is  a  common  asymptote  of  this  family  of  curves  since  if 

w  =  0 
then 

v  =  0 
and 

LdwJu=o~ 

The  line  (l/r/i,  —  rjifrji^),  the  reciprocal  of  the  asymptote  (171, 172)  of  the  system 
of  logarithmic  curves,  is  also  a  common  asymptote  of  this  family  of  curves. 

Subject  the  plane  to  the  transformation 

*  Motion  along  this  system  of  curves  corresponds  to  the  transformation 

Wo  =  otWo  +  i?(l  —a). 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


29 


v'  =  V  -\-  72. 
The  asymptote  (0,  0)  becomes  the  asymptote  (71,  72)  and,  if 

1 


and 


Vi 


+  7i  =  5i 


-  -i  +  72  =  02, 


the  asymptote  (Ifrji,  —  172/771^)  becomes  the  asymptote  (5i,  52).     The  equation 
of  the  system  of  curves,  c  being  an  arbitrary  constant,  is,  dropping  accents, 


oi2       r  (u  —  7i)(m  — 
«i  log  ai  L  Ti 


^^^^ ~     [log  (m  -  81)  -  log  (w  -  7i)] 

,    r_  c(7i  -  81)  +  (72  -  §2)   ,         (X2       log  (71  -  81)  "I    2 
L  (71  —  5i)^  «i  log  «i     (71  —  5i)     J 

rc(7i  —  5i)(7i  +  8x)  +  271(72  —  82)  _       <X2       (71  +  ^i)  log  (71  —  5i)"| 
L  (7i  —  5i)^  ai  log  ai  (71  —  61)  J 

_l_  r  (Ti  —  SiYjcyi  +  72)  —  C7i^(7i  —  ^1 
"^  L  (Ti  -  5i)' 


ai  log  ai 
)  -  Ti^(T2  -  82) 


I        «2       71^1  log  (Ti  —  5i)  1  ^ 
ai  log  ai  (71  —  81)         J  ■ 


ASYMPTOTE 


ASYMPTOTE 


Fig.  5. 

*  A  curve  with  this  equation,  ai/ai  log  ai,  71,  72,  Si,  82,  and  c  being  constants,  will  be 
referred  to  as  a  curve  (C).  In  Fig.  5  two  particular  examples  of  the  curve  (C)  are  given. 
By  a  system  of  curves  (C)  will  be  meant  the  « 1  curves  (C)  obtained  by  taking  definite  values 
for  the  constants  aijai  log  ai,  71,  72,  Si,  and  S2  and  letting  the  constant  c  be  arbitrary. 


30  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

The  equation  of  the  transformation  represented  by  motion  along  this  system 
of  curves  may  be  derived  in  the  following  way.  Suppose  the  lines  are  to 
move  from  the  initial  position  w  to  the  final  position  W.  If  the  plane  under- 
go the  transformation 

Wo  =    w  —  y, 

Wo=W-y, 

the  curves,  regarded  as  involutes,  will  be  unchanged  but  the  asymptote 

(Ti,  72)  will  become  the  asymptote  (0,  0).    As  lines  move  along  this  system  of 

curves,  with  one  asymptote  (0,  0),  from  the  positions  {wq)  to  the  positions 

(Wo),  their  reciprocal  lines  {wi  =  Ijwo)  and  (JVi  =  I /Wo)  will  move  according 

to  the  relation 

Wi  =  awi  +  /3. 

Hence 

1            1    ■    o 
— -  =  « \-^ 

Wo         Wo 


or 


and 


or 


Wo=       "° 


W-y  = 


j8wo  +  a 
w  —  y 


W  = 


p{w  —  7)  +  a 

(1  +  Py)w  +(ay-fiy^-y) 
fiw+{a-  ^y) 


The  asymptote  5  may  be  introduced  instead  of  the  quantity  /3.  The 
transformation  Wo  =  w  —  y  changes  the  line  5  into  the  line  8  —  y.  The 
motion 

Wi  =  awi  +  jS 

takes  place  about  the  fixed  line  /3/(l  —  a)  which  is  the  reciprocal  of  the  line 
5  —  7.    Hence 

/3      _      1 
1  —  a      8  —  y 
or 

1  -a 
8  —  y 

and  the  transformation  takes  the  form 

^  (5  -  ay)w  +  icx8y  —  8y) 
(1  —a)w  +(a5  —  7)      * 

The  lines  7  and  5  are  the  asymptotes  of  the  system  of  curves  (C)  and  a  defines 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  31 

the  extent  of  the  motion  which  is  reciprocal  to  the  given  motion  with  respect 
to  the  asymptote. 

The  transformation  may  be  written 

8(w  —  y)  —  ay{w  —  S) 
{w  —  y)  —  a{w  —  5)    * 

The  reverse  transformation  is  then 

^  y{W  -8)-  a8iW  -  y) 
^  ~     (W-  8)-  (x{W  -  y)    ' 

If  the  first  of  these  moves  w  to  W,  the  second  moves  W  back  to  w.  Each 
transformation  may  be  obtained  from  the  other  by  interchanging  y  and  5. 
Hence  if  the  asymptotes  of  the  motion  be  interchanged,  a  second  motion  equal 
and  opposite  to  the  first  will  be  obtained. 

The  transformation  is  the  general  linear  transformation  and  may  be  put 
in  the  form 

y'w  +  5' 
or,  dropping  accents, 

aw  +  j8 


W  = 


yw  +  5 


{d)   The  Fixed  Lines  and  the  Conditions  for  the  Types  of  Motion. 

It  has  been  shown  that  motions  along  the  several  systems  of  curves  corre- 
spond to  linear  transformations  of  the  variable  u  +  jv.  It  will  now  be  shown 
that  all  linear  transformations  give  rise  to  the  same  systems  of  curves. 

The  transformation 

aw  +  j3 


w  ^ 


yw  +  5 


in  general  leaves  unchanged  two  Hues  of  the  plane  which  are  the  roots  of  the 
equation 

yw^  +  (5  —  a)w  —  j8  =  0. 

If  the  two  roots  are  coincident  and  infinite,  then 

7  =  0 
and 

6  -  a  =  0 

and  the  transformation  reduces  to 

W  =  W  +  -. 
a 

If  the  transformation  is  of  this  form,  then  it  has  no  fixed  lines  in  the  finite  plane. 


32  CONTWBUnONS  TO  EQUILONG  GEOMETRY. 

If  the  two  roots  are  coincident  and  finite,  then 

(5  -  a)2  +  4/37  =  0 

and  the  transformation  reduces  to 

(8  -  a)« 

aw  —  — -. 

w=  ^ 


8) 

(a  -  dy  +  4i87  = 

=  0 

[K«  +  8)f 

1 

yw  +  5        * 

the  motion  corresponding  to  which  has  been  called  parabolic.     A  transforma- 
tion of  this  form  has  two  coincident  lines  fixed  in  the  finite  plane. 
The  fixed  lines  ivi  and  W2>  for  parabolic  motion,  are 


The  condition 

may  be  written 

! 

a8  —  fiy 

If  the  motion  takes  place  along  points  at  infinity,  then,  from  the  equation 
for  parabolic  motion  in  the  form 

{\±^y)w±S-M 
^w+{i-  /St)      * 

^  =    0  +  i^2. 

If 

b  +  ^by  =  a', 

^h  =  7', 
and 

b-phy=  b\ 
then 

„  _     2y'      _     (g/  +  W)yx'  +  j[{ax'  +  8i072^  -  («2^  +  8207/1 
^~a'  +  b'      ^  («i'  +  V)^ 

The  further  condition  that  parabolic  motion  take  place  along  points  at  in- 
finity is  then,  dropping  accents,  that,  in  the  equation 

(8  -  af 
aw ^, 

W  =  -rP^—  ,  71  =  0. 

yw  +  0 

If  there  is  only  one  fixed  line  in  the  finite  plane,  then 

7  =  0. 
The  transformation  is 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  33 

a  8 

W^-^w  +  l. 

Conversely,  this  transformation  has  one  fixed  line  j8/(5  —  a),  which  is  the 
asymptote  of  the  system  of  logarithmic  curves. 

If  the  two  fixed  lines  are  distinct  and  in  the  finite  plane,  one  may  be  made 
to  become  the  line  (0,  0)  by  the  rotation-dilatation  transformation,  the  other 
becoming  at  the  same  time  some  other  line  in  the  plane.  By  reciprocation, 
the  line  not  (0,  0)  is  transformed  to  some  part  of  the  plane.  Motion  about 
one  line  in  the  finite  plane  is  motion  along  a  system  of  logarithmic  curves. 
Suppose  the  operations  be  reversed.  The  reciprocal  of  the  motion  along  the 
system  of  logarithmic  curves  is  motion  along  a  system  of  curves  (C)  which 
has  the  line  (0,  0)  and  the  fine  into  which  the  second  fixed  line  of  the  original 
transformation  was  transformed  as  asymptotes.  Reversing  the  rotation- 
dilatation  transformation,  motion  along  the  system  of  curves  (C)  which  has 
the  two  distinct  fixed  lines  in  the  finite  plane  as  asymptotes  is  obtained.  In 
general,  the  linear  transformation  which  leaves  two  distinct  lines  fixed  in  the 
finite  plane  is  represented  by  motion  along  a  system  of  curves  (C).  The 
fixed  lines,  Wi  and  1^2,  of  this  motion  are 


^(«-5)+V[^(«-5)P4-^7 
Wi  = 


and 

2 


nh  = 


Hoc  -  5)  -  V[i(«-5)P  +  ^7 


To  determine  the  conditions  for  the  hyperbolic  case,  the  equation  of  the 
transformation  in  terms  of  the  fixed  lines  and  the  quantity  a,  now  called  €, 

_  (vh  —  iWi)w  +  WitOije  —  1) 

(1  —  e)W  +  €W2  —  Wi  ' 

is  used.    If  this  transformation  is  identical  with  the  transformation 

aw  +  /3 


then,  fc  being  a  constant. 


Then 


yw  -\-  0 

k(w2  —  ewi)  =  a, 

kwiuhie  —  1)  =  /3, 

kil  -  6)  =  7, 

k{€W2  —  Wi)  =   8. 

W2  —  €Wi  _  a 


34  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

and 

,    .         a  +  8  +  V(a  +  5)2  -  4(a5  -  py) 

€  =   Cl  +  762  =    , 

a  +  5  -  V(a  +  5)2  -  4(a5  -  ^) 


If 
then 

and 


[{a  +  8)  +  -^(«  +  5)2  -  4(a5  -  ^y)Y 
4(a5  -  py) 

€2   =   0, 


oci8i  —  j8i7i 


>  1 


{fXi  +  52)(q;i5i  —  /3i7i)  —  §(q:i  +  5i)(ai52  +  o;25i  —  /SoTi  —  /3i72)  =  0. 
The  condition  for  hyperbolic  motion  is  then  that,  in  the  transformation 

yw  -\-  5 ' 
[hicc  +  5)]2 


ab  —  fiy 


>  1. 


For  the  case  in  which  the  motion  is  along  points  at  infinity,  the  form  of 
the  transformation 

^  (1  +  ^y)w  +  («T  -  fiy^  -  y) 

is  used.     If  this  transformation  is  identical  with  the  transformation 

^       y'w-{-8" 
then 

8(1  +  /St)  =  «', 

diay  -  ^7'  -  7)  =  ^', 

8^  =  7', 

8(a  -  ^7)  =  8', 
and 

a  -  1         a'8'  -  ^'y' 

(a  +  1)^  ~    («'  +  8'f  • 
If  a  is  of  the  form  1  +  ja^,  then 

•   _  ^'^'  -  ^'y' 

For  the  motion  along  points  at  infinity  the  condition  is,  dropping  accents 
and  thereby  taking  the  transformation  to  be 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  35 


yw  -j-  5' 
that 

a8  —  ^y 

contains  j. 

The  general  transformation 

aw  -\-  0 


W  = 


yw  +  5 


contains  three  essential  constants  of  the  form  771  -]r  j'fh-  There  are  then  00" 
such  real  transformations. 

In  the  general  transformation,  four  constants  determine  the  asymptotes, 
one  constant  determines  the  system  of  curves  (C),  and  one  constant  deter- 
mines the  extent  of  the  motion  on  the  system. 

One  condition  determines  hyperbolic  motion.  Four  constants  determine 
the  system  of  curves  and  one  constant  determines  the  extent  of  the  motion. 

Two  conditions  are  required  for  parabolic  motion.  Two  constants  deter- 
mine the  common  element  of  the  system  of  second  involutes  of  circles,  one 
constant  determines  the  distance  to  the  common  point  of  tangency,  and  one 
constant  determines  the  extent  of  the  motion. 

All  linear  transformations,  being  obtainable  from  combinations  of  the 
transformations  W  =  w  -\-  ^,  W  =  aw,  and  W  =  Ifw,  of  course  possess  the 
equilong  property. 

{e)  The  Simplest  Invariant  of  a  Regular  Analytic  Curve  under  the  General 

Linear  Transformation. 

The  invariant  of  lowest  order  for  the  general  linear  transformation  will 
next  be  derived.  All  the  linear  transformations  which  leave  the  line  (0,  0) 
unchanged  are  represented  by  the  equation 

W  = ; — :  =  aw(l  +  yw)~^. 

yw  +  I 

Expanding  the  right-hand  member,  the  transformations  may  be  written 
U=  Eai(-7i)'"w'"+S 

00 

F  =  Z)  Hn  +  l)ai{-  7i)"  -  (nai(-  T^^'^Ta  -  «2(-  yi)'')u]u''. 

n=0 

If  such  a  transformation  turn  the  analytic  curve 

00 
V  =  ^  hkU^ 


36  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 

into  the  analytic  curve 

k=l 

then  the  result  obtained  by  eliminating  TJ,  V,  and  v  among  the  four  equations 
is  an  identity  with  respect  to  u  or 

Z  I  E  h^in  +  l)ai(-  7i)«  -  (p  -  l)ai(-  71)^^2  +  a^{-  7i)^' 

p=l    I   n=0 

-  £  5^1'     /'        (-  y.f'^  .  •  •  (-  Ti)*^  1  u^  =  0, 

a=l  XqI'  '  'Xtl  J 

a?o.  •  •  • ,  art  being  positive  integers  or  zero  satisfying  the  relations 

Xi  +  2«2  +    "  '  tXt  =  p  —  Sy 

Xq-\-  Xi-\-  "  '    Xt  =  s. 

If  the  coefficients  of  the  first  five  powers  of  u  be  each  equated  to  zero,  the 
constants  (oci,  a^,  71,  72)  of  the  transformation  may  be  ehminated  with  the 
result 

W-hh      B^-BzB,' 

The  left  hand  member  of  this  equation  is  therefore  invariant  under  the  trans- 

00 

formations.     In  terms  of  the  derivatives  of  the  curve  v  =  ^  hkUk  at  the 

*=i 
line  (0,  0),  this  invariant  has  the  form 

3 


J5  = 


/^Y_   /^\/d^y 


The  transformations  IF  =  «)  +  j3  and  W  =  1/w  are  not  included  in  the 
transformations  W  =  aw{l  +  7^)"^  For  the  transformations  W  =  w  -\r  ^ 
and  W  =  1/w,  direct  computation  shows  that  J5  is  invariant. 

Hence  the  invariant  of  lowest  order  for  the  general  linear  transformation 
W  =  (aM>  +  /3)(7W  +  6)-i  is 


^6  = 


(dhV 
\du^) 


/^Y_     /^Wrf^y 


A  family  of  curves  characterized  by  the  property 

Js  =  T  (a  constant) 


CONTRIBUTIONS  TO  EQUILONG  GEOMETRY.  37 

is  invariant  under  the  general  linear  transformation.     By  solving  the  dif- 
ferential equation 

\du')  ^  1 

the  00  ^  curves  of  the  family  are  found  to  be 


v=  — 
2k 


;^[(..+  6)2  +  ^][log(i.+  6-^AP^)-log(i^+64-^V-A:)] 


a 
a,  h,  c,  d,  and  e  being  arbitrary  constants. 


+  i§-^^  +  ^[d+^]i.  +  e, 


38  CONTRIBUTIONS  TO  EQUILONG  GEOMETRY. 


VITA. 

Paul  Henry  Linehan  was  born  in  Boston,  Massachusetts,  January  15, 
1879.  He  attended  the  public  schools  of  his  native  city  and  Harvard  Uni- 
versity, from  which  he  received  the  degree  of  Bachelor  of  Arts  in  1902. 
Since  February,  1903,  he  has  been,  successively,  tutor  and  instructor  in 
mathematics  in  the  College  of  the  City  of  New  York. 


L5- 


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